The turtle follows the path of the Hilbert curve of some order < 10. What cool generative art could be generated by Hilbert curves besides walking through rgb-colors with it? I thought about generating 2D perlin-like-noise from 1D perlin-noise with it.

Procedurally generated video of stars and clouds moving using a parallax effect.

This video actually consists of multiple pieces of procedurally generated art combined together. These individual parts are the star and cloud textures, the code that animates them and the Bézier curve gradients that add colour.

I created this art as a part of Spaceships, a free/libre video game that I created. The game and this art are created in the Rust programming language and using the glium crate for rendering the video (this is a safe abstraction to OpenGL).

Each individual star texture is generated from four glowing lines that intersect the centre of the texture. I was inspired by GIMP's sparkle plugin on individual pixels when making this.

The clouds are generated using Perlin noise in a similar way to how this article ge

We call two numbers x,y coprime if their greatest common divisor is 1.

Examples

• 9 and 8 are coprime
• 3 and 7 are coprime. gcd(3, 7) = 1
• 11 and 99 are not coprime, since gcd(99,11) = 11 != 1

Lower Diagonal y>x

A pixel x,y is marked black, if x and y are coprime.

Upper Diagonal x>y

A pixel x,y is marked black, if 2x+1 and 2y+1 are coprime

The different behavior for the upper/lower diagonal was chosen, since gcd is commotative, and the result would have been a boring mirror image.

generated with the following c-code:

 c

    
#include "intmaths.h"
#include <stdio.h>
#include <assert.h>

int main(void){
  // tests to check if gcd works
  assert(3 == gcd(3*5, 3*7));
  assert(11 == gcd(11*5, 11*7));
  assert(1 == is_prime(3));
  int t1 = gcd(11*3*3, 3*7);
  assert(t1 == 3);
  int t2 = gcd(11*4, 4*7);
  assert(t2 == 4);
  int W = 300;
  int H = 300;
  int START = 2;
  printf("P1\n%d %d\n", W-START, H-START);
 
  

Today we want to generate this star like shape using sums of trigeometric functions sin and cos:

Function:

 undefined

    
f(x) = x/57 + x**3/19 

  

where x**3 is x^3 = x*x*x written in python

To calculate the x and y coordinate of the nth. step:

 undefined

    
sx(n) = sum((75*cos(2*pi*f(i)) for i in range(n)))
sy(n) = sum((75*sin(2*pi*f(i)) for i in range(n)))

  

To render this with pythons turtle library, the following code can be used.

 python

    
from math import cos, sin, pi, tan
def f(x):
    form = x/57 + x**3/19
    return form

def seq(fu):
    r = 75 # "zoom" level, kinda arbitrary choice so you can see it well
    s = [0, 0]
    for i in range(10000):
        s[0] += r*cos(2*pi*fu(i))
        s[1] += r*sin(2*pi*fu(i))
        yield s

import turtle
from time import sleep
for i in seq(f):
    turtle.setpos(i[0], i[1])
sleep(20)

  

This expone